주어진 수보다 작은 소수의 개수에 관하여

〈주어진 수보다 작은 소수의 개수에 관하여〉(독일어: Über die Anzahl der Primzahlen unter einer gegebenen Größe 위버 디 안찰 데르 프림찰렌 운터 아이너 게게베넨 그뢰세[*])는 베른하르트 리만이 1859년 11월에 베를린 학술원에 발표한 8페이지짜리의 독창적인 논문이다.[1] 비록 이것은 리만이 발표한 수론에 관한 유일한 논문이지만, 19세기말부터 현재에 이르기까지 수많은 연구자들에 영향을 미치고 있는 개념들을 포함하고 있다. 또 이 논문은 증명들을 위한 밑그림들과 발견을 돕는 논증들과 정의들, 그리고 강력한 해석학적 방법론의 응용으로 이루어져 있다. 수학적 증명의 난제로 알려진 리만 가설도 이 논문에 나온다. 이 모든 것들은 근대 해석적 수론의 필수적인 개념과 도구들이 되었다.

내용

논문 중의 새로운 정의들은 소개하면 아래와 같다:

  • 이전에 오일러가 언급하였던 함수에 대한 그리스 문자 제타의 사용
  • s = 1를 제외한 모든 복소수에서 리만 제타 함수 ζ(s)의 해석적 연속
  • 감마 함수를 통해 제타 함수와 관련된 전해석 함수 ξ(s) 정의
  • 이고, 소수의 거듭제곱 에서 씩 증가하는 양의 실수에서 정의된 이산 함수(discrete function) .

다음 증명들을 제시했다.

  • 의 함수 방정식의 두 가지 증명법
  • 의 무한 곱 형태 표현(일명 의 '인수분해')의 증명
  • 허수부가 0과 사이인 임계띠 안의 의 근의 개수의 근사식의 증명

다음 추측을 제시했다.

수론에서 다음 새로운 기법을 도입했다.

각주

  1. ↑ 주어진 수보다 작은 소수의 개수에 관하여, 베른하르트 리만, 영문판

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